In the stamping industry, shape distortion due to springback from flanging operations is a serious problem. In many cases, manual correction for shape distortion remains the common practice. It is estimated that $100 million dollars is spent each year by North American stamping plants alone to correct shape distortion defects. This concern is even more critical for lightweight materials such as high-strength steel and aluminum. The generation of springback is based on the hardening rule in the mathematical theory of plasticity used in sheet metal forming simulations.
For simplicity, most sheet metal forming simulation codes use the isotropic hardening rule developed by the mathematical theory of plasticity. However, this hardening rule does not produce realistic numerical results when used to analyze cyclic loading and unloading processes, such as those associated with stretching, bending and unbending over a small radius, or straightening an initial wrinkling as happens in a draw forming operation. FIG. 1 shows that the amplitudes of the uniaxial stress under the specified strain history predicted by the isotropic hardening rule are unreasonably high. Therefore, an anisotropic hardening theory to establish the incremental stress/strain relationship should be used for a more exact simulation of sheet metal forming processes.
The simplest anisotropic hardening rule is the kinematic rule by Prager and Ziegler. This hardening rule has been used to simulate the Bauschinger effect. However, for complex loading histories, actual material behavior substantially deviates from that predicted by this kinematic hardening rule. In addition, there is no definite method for determining the tangent modulus for a non-linear hardening material. The hardening rule proposed by Mroz, "On the description of anisotropic work hardening; J. MECH. PHYS. SOLIDS, Vol. 15, pp. 163-175, 1967, based on an observation of material fatigue behavior, is more appropriate for studying the influence of complex loading histories on material behavior which cannot be explained by either the isotropic or the kinematic hardening rule.
FIG. 2 shows the amplitudes of the uniaxial stress predicted by Mroz's rule under the same strain history as that in FIG. 1. Notably, the stress up to A' is identical to that by the kinematic hardening rule for the uniaxial stress case; however, there is no difficulty in determining the tangent modulus for a nonlinear hardening material.
C. Chu in "A three dimensional model of anisotropic hardening in metals and its application to the analysis of sheet metal formability," J. MECH. PHYS. SOLIDS, Vol. 32, pp. 197-212, 1984 extended Mroz's rule to establish a general constitutive equation in terms of the Cartesian tensor for the elastic plastic material in a three dimensional continuum. Unlike the isotropic and kinematic hardening rules, in which a single yield surface is assumed either to expand or translate, respectively, as a result of plastic deformation, Mroz's model introduced the concept of the field of work-hardening moduli which are defined by the configuration of a finite number of initially concentric yield surfaces in deviatoric stress space. The general rules governing the configuration change are that yield surfaces must move as rigid bodies with the loading point when it is in contact with them, and that the surfaces cannot pass through one another. Therefore, the surfaces will become mutually tangent at the loading point. As it passes from the elastic into the plastic region, the loading point will first encounter the smallest yield surface, which has a radius .sqroot.2/3.sigma..sub.0 wherein .sigma..sub.0 is the initial yield stress. This surface will be pushed forward until the next larger surface is reached and then these two surfaces will then move forward together and so on. Each of these yield surfaces has a constant work hardening modulus. Since a surface is permitted only rigid body motion, its size may be used as a parameter to determine the modulus. When the material is deep into the plastic range and there are multiple yield surfaces tangent to one another at the loading point, the instantaneous modulus for continuous loading is the one associated with the largest yield surface in contact. This is the active yield surface. The smaller yield surfaces can become active again whenever unloading and reloading takes place.
The following derives the equations for change-of-yield surface size and central movement. The Von Mises yield criterion in Cartesian coordinate system is used. According to Mroz's rule, the yield's function is written as: EQU r=(3/2) (s.sub.ij -a.sub.ij) (s.sub.ij -a.sub.ij)-k.sup.2 =0(i,j=1,3)(1)
where s.sub.ij are the deviatoric components of the Cauchy stress tensor .SIGMA.a is the position tensor of the center of the active yield surface, and .sqroot.2/3 k is the radius of this surface. Note that the boldface character denotes a tensor, the index denotes its component and the repeated index means summation. The differential form of the yield's function is EQU (3/2) (s.sub.ij -a.sub.ij) (ds.sub.ij -da.sub.ij)-kdk=0 (2)
assuming the yield surface moves along a unit tensor b, the magnitude of da is the increment of the radius of the yield surface. Therefore, ##EQU1## Substituting this equation into Equation 2 yields EQU dk=(3/2) (s.sub.ij -a.sub.ij)ds.sub.ij /k (4)
where ##EQU2## and Equation 3 becomes ##EQU3## If the associated flow rule is assumed, the elastic, plastic constitutive equation can be derived in a procedure similar to that by means of the isotropic hardening rule.
An example depicting the change of active yield surfaces in a process for initial loading, unloading, reloading, unloading again and then reloading in a multiple-dimensional deviatoric stress component space is illustrated as follows:
1. Initial and Continuous Loading. The center of the initial yield surface is at the origin and its radius is .sqroot.2/3.sigma..sub.0 as shown in FIG. 3. One continuously loads to point A.sub.o where the radius of the yield surface is .sqroot.2/3k as shown in FIG. 3. The unit tensor b in Equation (3) is along OA.sub.o. The center of the smallest yield surface with radius .sqroot.2/3.sigma..sub.0 go moves to 0.sub.1.sup.(1).
2. Unloading and Reloading. One unloads inside the smallest yield surface with center at 0.sub.1.sup.(1) and reloads to A.sub.1 with the deviatoric stress increment ds. Using this increment and the unit tensor b, one can compute the center 0.sub.1.sup.(i) of the bigger yield surface and its radius .sqroot.2/3 k.sub.1 from Equations (4) and (5). The updated unit tensor b is along 0.sub.1.sup.(i) A.sub.1 and the center of the updated smallest yield surface is along this line and thus can be located at 0.sub.2.sup.(1) as shown in FIG. 3.
3. Unloading Again and Then Reloading. If one unloads again inside the newest smallest yield surface and reloads into the plastic range again, the center of the active yield surface is along the line 0.sub.1.sup.(i) 0.sub.2.sup.(1). This center cannot go beyond 0.sub.1.sup.(i) as shown in FIG. 3. If it does, the new center will lie on the line OA.sub.o. If continuous loading occurs, the center of the active yield surface cannot go beyond O; otherwise, the yield surface with the radius .sqroot.2/3 k will be active and move to tangent a yield surface with bigger radius than .sqroot.2/3 k and its center still at 0.
The hardening rule proposed by Mroz better explores the influence of complex loading histories on material behavior which cannot be explained by either the isotropic or the kinematic hardening rule. However, Mroz's model cannot accurately predict springback for nonlinear hardening materials. The linear elastic material model underestimates the amount of springback, while the isotropic hardening rule makes wrong predictions.
Accordingly, there is a need for a revised approach to the traditional isotropic hardening rule, one which allows a more accurate simulation of forming processes and particularly the prediction of springback for nonlinear materials.